A recent discussion in [1]} highlights that a source covariance \(\Sigma _{\mathrm {q}}\)  that penalizes the equation error away from the point-source position produces a better-conditioned problem for inversion. In [2]}, for example, \(\Sigma _{\mathrm {q}}=\mathrm {diag}(\mathbf {\sigma }^2)\)  is a diagonal matrix defined as the inverse of the squared distance from the source location. We exploit the “source-focusing” character of this type of covariances by adopting a stochastic low-rank approximation:
\(\Sigma _{\mathrm {q}}=\mathrm {diag}(\mathbf {\sigma }^2)=\mathbb {E}_{\mathbf {z}}\ \mathbf {z}\mathbf {z}^*,\ \ \ \mathbf {z}\sim \mathcal {N}(\mathbf {0},\mathrm {diag}\mathbf {\sigma })\quad \Rightarrow \quad \mathcal {L}(\mathbf {m},\mathbf {y})=\mathbb {E}_{\mathbf {z}}\ \mathcal {L}(\mathbf {m},\mathbf {y};\mathbf {z}),\\\mathcal {L}(\mathbf {m},\mathbf {y};\mathbf {z})=\langle \mathbf {y},\mathbf {r}(\mathbf {m})\rangle -\dfrac{1}{2}\left\Vert \mathbf {y}\right\Vert ^2_{\widetilde{\Sigma }_{\mathrm {d}}(\mathbf {m};\mathbf {z})^{-1}},\quad \widetilde{\Sigma }_{\mathrm {d}}(\mathbf {m};\mathbf {z})=\Sigma _{\mathrm {d}}+\mathbf {d}_{\mathbf {z}}\mathbf {d}_{\mathbf {z}}{}^*, \ \mathbf {d}_{\mathbf {z}}=\mathrm {F}(\mathbf {m})\mathbf {z}.\) 
